There usually is very little snow there by the end of June. I would say the trails are usually snow free by about the end of May. But we had a crazy winter here in So NV in 2005 with record rains in Vegas and huge snows in the mountains.
After hiking Mt. Charleston countless times in warm weather, I did my first winter climb up through the Big Falls area last April. I'm hooked and can't wait for the snow to come back so I can do it again!